Non division dans \(A^{\infty}(\Omega)\)
Let \(\Omega\) be a pseudoconvex domain, bounded with \(C^{\infty}\) smooth boundary and let \(A^{\infty}(\Omega)\) be the set of holomorphic functions in \(\Omega\), \(C^{\infty}\) smooth up to the boundary of \(\Omega\). Let X be a holomorphic subvariety of \(\Omega\) defined by: \[ (*)\quad X=\{z\in {\bar\Omega }u_ 1(z)=...=u_ k(z)=0\} \] with: \[ (**)\quad u_ i\in A^{\infty}(\Omega),i=1,2,...,k\quad and\partial u_ 1\cap...\cap\partial u_ k\neq 0on\quad X\cap\partial \Omega. \] Now if f is in \(A^{\infty}(\Omega)\), f is zero on X, the question of the factorisation of f arises: does exist \(f_ 1,...,f_ k\) in \(A^{\infty}(\Omega)\) such that: \(f=\sum^{k}_{i=1}f_ iu_ i?\) The answer is yes if (1) X and \(\partial\Omega \) have a bounded order of contact [the author, ''\({\bar\partial }\)-cohomologie \(C^{\infty}\) et applications'' (1980)]; (2) X and \(\partial\Omega \) are regularly separated [\textit{P. de Bartolomeis} and \textit{G. Tomassini}, Adv. Math. 46, 162-170 (1982; Zbl 0499.32012), in the case \(\Omega\) strictly pseudoconvex, but with a more general X). In this work we prove that if X and \(\partial\Omega \) are not regularly separated, then there is no factorisation in \(A^{\infty}(\Omega)\). More precisely, if \(\tilde X\) is a \(C^{\infty}\) Whitney extension of X in a neighborhood of \({\bar\Omega }\) we have: Theorem: Let \(\Omega\) be a bounded pseudoconvex domain in \({\mathbb{C}}^ n\) with \(C^{\infty}\) smooth boundary and let X be a holomorphic subvariety satisfying (*) and (**), and such that: (\(\alpha)\) \({\bar\Omega }\) and \(\bar{\tilde X}\setminus {\bar\Omega}\) are not regularly separated at a point \(p\in\partial \Omega\cap X\); (\(\beta)\) p is a point of strict pseudoconvexity of \(\partial\Omega \); then there is an f in \(A^{\infty}(\Omega)\), f is zero on \(X\cap {\bar\Omega }\) but f cannot be factorised in \(A^{\infty}(\Omega)\). - Corollary. Let \(\Omega\) be a strictly pseudoconvex domain in \({\mathbb{C}}^ n\), bounded with smooth boundary and let X be a holomorphic subvariety satisfying (*) and (**), then (\(\alpha)\) and (\(\beta)\) are equivalent: (\(\alpha)\) any function in \(A^{\infty}(\Omega)\), zero on X can be factorised in \(A^{\infty}(\Omega)\); (\(\beta)\) \({\bar\Omega}\) and \(\bar{\tilde X}\setminus {\bar\Omega}\) are regularly separated.
no factorization in algebra of holomorphic functions
scientific article; zbMATH DE number 3871765